The Intersection Of A₁-subgroups Of Finite ρ-groups

Suppose that G is a finite p-group. If all subgroups of index pt of G are abelian and at least one subgroup of index pt-1of G is not abelian, then G is called an At group. In this paper, we classify finite p-groups whose intersection of any two A1-subgroups is a maximal subgroup of any one of them; in addition, we also classify finite p-groups whose any two different A1-subgroups generate a A2-subgroup.